3 1 t 2 dy dt 2ty y 3 1

Applied Differential Equations

Name:

• I will be checking for organization, conceptual understanding, and proper mathematical communication, as well as completion of the problems.

• Show as much work as you can, draw sketches if necessary and clearly explain why you are doing what you are doing. • Use correct mathematical notation. • You may work with your classmates. However, please submit your own work! • Please use this sheet ONLY as a cover page. Work on a separate sheet of paper.

1. Consider the following IVP: dy

dt = 2ty, y(0) = 1 (1)

(b) (3 points) Solve the IVP given in (1) analytically. Use your solution to determine y(1). Round your answer to four decimal places.

(c) (1 point) Compute the percent error:

Percent error = |Actual value−Approximate value|

Actual value × 100%.

(d) (1 point) Comment on the result obtained in part (c). How good was your approximation? How could you improve the accuracy of your approximation?

2. Harvesting a Renewable Resource Suppose that the population P of a certain species of fish in a given area of the ocean is described by the logistic equation (please read “Notes for Section 1.1”)

dP

dt = k

( 1− P

N

) P.

If the population is subjected to harvesting at a rate H(P, t) members per month, then the harvested population is modeled by the differential equation

dP

dt = k

( 1− P

N

) P −H(P, t) . (2)

Although it is desirable to utilize the fish as a food source, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level and possibly even driven to extinction.

(OVER)

1

(a) (4 points) Use Euler’s method with the step size ∆t = 0.1 to approximate the solution to (1) over the time interval 0 ≤ t ≤ 1 and determine the value of y(1). Your answer should include a table similar to the table in Section 1.4 Examples 1 or “HW#3: Section 1.4 - Problems 6 or 8.” Only display four decimal places but make sure to keep ALL digits for successive calculations. It should also include a sketch of the graph of the approximate solution.

The following problems explore some of the questions involved in formulating a rational strategy for managing the fishery!

Constant Effort Harvesting At a given level of effort, it is reasonable to assume that the rate at which fish are caught depends on the population P : the more fish there are, the easier it is to catch them. Thus we assume that the rate at which fish are caught is given by H(P, t) = EP , where E is a positive constant, with units of ‘1/month’, that measures the total effort made to harvest the given species of fish. With this choice for H(P, t), Equation (2) becomes

dP

dt = k

( 1− P

N

) P − EP . (3)

(This equation is known as the Schaefer model after the biologist M.B. Schaefer.)

(a) (2 point) Show that there are two equilibrium points, P1 = 0 and P2 = N(1 − E/k). [Note: you need to actually solve for these, rather than showing the derivatives of P1 and P2 have a derivative that is 0. This way you you’re showing there are only two equilibrium solutions.]

(b) (2 points) Assume E < k. Sketch the phase line for P ≥ 0. Also, classify the equilibrium point (source, sink, or node) only for P2.

(c) (2 point) Sketch several graphs of solutions.

3. (5 points) Consider the autonomous differential equation below. Locate the bifurcation value for the one- parameter family and draw the phase lines for values of the parameter slightly smaller than, slightly larger than, and at the bifurcation value.

dy